Combine the milk with the sugar and salt in a medium-sized pot over medium heat. Whisk until the mixture is hot to touch, and the sugar has completely dissolved.

Whisk together the egg yolks and cornstarch.

While stirring, gradually pour ~½ cup of the milk mixture into the yolk mixture. Then, add the yolk mixture to the pan with the remaining milk mixture.

With the pan over medium heat, stir until the pudding is thick and bubbly. Remove from heat and whisk in the vanilla extract and softened butter until it's melted and completely combined.

Cover the top of the pudding with plastic wrap and let it cool.